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        Questions tagged [lie-groups]

        Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

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        votes
        0answers
        30 views

        First homology group of the general linear group

        The abelianization of the general linear group $GL(n,\mathbb{R})$ defined by $GL(n)^{ab} := GL(n)/[GL(n), GL(n)]$ is isomorphic to $\mathbb{R}^{\times}$. This follows from the fact that $[GL(n),GL(n)] ...
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        votes
        0answers
        52 views

        Topological derivation of the Laplace formula for determinants in Euclidean space

        I'm interested in trying to derive the Leibniz formula for the determinant on real Euclidean space, without first constructing $\det$ by axiomatizing its properties. We know $GL(n)$ deformation ...
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        votes
        1answer
        70 views

        Reference Request: Carnot Group Not Containing Group of Isometries

        This question is a follow-up to this post, from which I quote: Let $\mathfrak{e}$ be the 3-dimensional Lie algebra with basis $(H,X,Y)$ and bracket $[H,X]=Y$, $[H,Y]=-X$, $[X,Y]=0$. It is ...
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        votes
        1answer
        242 views

        Lie Algebra of Automorphism Group of $\mathbb{P}_k^1$

        Let $X$ be a scheme over an algebraically closed field $k$ and let $\operatorname{Aut}(X)$ denote the functor sending a $k$-scheme $T$ to the group $\operatorname{Aut}_T(X \times_k T)$ of ...
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        votes
        0answers
        206 views

        Interpretation of determinants on commutative rings

        In real Euclidian space, the result of the determinant can be interpreted as the oriented volume of the image of the unit cube under an invertible linear map. This interpretation conceptually depends ...
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        votes
        1answer
        219 views

        History of the notion of $(G,X)$-structure

        I'm currently searching for sources and historical basis on the notion of $(G,X)$-structure as it appears in Thurston's work. So far, it appears that he was the first to set it. Many mathematicans ...
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        56 views

        Lattices are not solvable in non-compact semisimple Lie groups

        I'm trying to prove the following result. If $G$ is a non compact semisimple Lie group with no compact factors (lying in some $SL(l,\mathbb{R})$), and $\Gamma$ is a lattice in $G$, then $\Gamma$ is ...
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        votes
        1answer
        103 views

        Computing Deligne-Lusztig Characters in General

        The goal for this question is to try to find a relatively explicit way of computing the Deligne-Lusztig characters. I understand that the $R_{T,\theta}$ can be computed if we know the values of the ...
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        votes
        0answers
        41 views

        Generalizing polynomial identities for rings

        For a ring $R$, a polynomial identity of $R$ is a polynomial (in non-commuting variables) $f(x_1,\ldots,x_n)\in \mathbb{Z}[x_1,\ldots, x_n]$ such that for any choice of $a_i\in R$, $f(a_1,\ldots, a_n)=...
        7
        votes
        1answer
        217 views

        Independence of Duistermaat-Heckman measure

        Suppose that a compact K?hler manifold $(X,\omega)$ has a real torus acting on it by symplectomorphisms in a Hamiltonian way (the torus is not necessarily of maximal rank). Then for any smooth ...
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        27 views

        Reference Request: Carnot Groups over Complexes

        Is there a theory of complex (analytic) Carnot groups and Caratheodory metrics?
        2
        votes
        1answer
        120 views

        On a criterion for rational-smoothness of Schubert varieties and an ambiguity of the taking the ambient Algebraic group to be simply connected or not

        In the paper: Pattern Avoidance and Rational Smoothness of Schubert Varieties, Sara C. Billey, Advances in Mathematics 139, 141-156(1998), https://www.sciencedirect.com/science/article/pii/...
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        votes
        3answers
        460 views

        Nonnegativity of an integral over the unitary group

        For an $n$-by-$n$ unitary matrix $U$ and a permutation $\sigma\in S_n$, let $$w_\sigma=(-1)^\sigma\det(U^*)\prod_{i=1}^n U_{i,\sigma(i)}.$$ Is $\int_{U(n)}\mathrm{Re}(w_{\sigma_1})\mathrm{Re}(w_{\...
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        vote
        0answers
        46 views

        Uniqueness of Equivariant Harmonic Map for Surface Group Representation

        In section 1.2 of https://arxiv.org/pdf/1311.2919.pdf the following result is stated. $\textbf{Theorem}$ (Labourie). Let $S$ be a closed Riemann surface of negative Euler characteristic, $Γ$ its ...
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        votes
        1answer
        160 views

        On some notations and notions of a paper on smoothness of Schubert varieties by Lakshmibai and Sandhya

        I am reading the paper Criterion for smoothness of Schubert varieties in $\mathrm{Sl}(n)/B$ by V Lakshmibai and B Sandhya; Proc. Indian Acad. Sci. (Math. Sci.), Vol. 100, No. 1, April 1990, pp. 45-52. ...

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